Finite nonassociative algebras obtained from skew polynomials and possible applications to (f, σ, δ)-codes
S. Pumplün · Advances in Mathematics of Communications · 2017
Let $S$ be a unital ring, $S[t;\sigma,\delta]$ a skew polynomial ring where $\sigma$ is an injective endomorphism and $\delta$ a left $\sigma$-derivation, and suppose $f\in S[t;\sigma,\delta]$ has degree $m$ and an invertible leading coefficient. Using right division by $f$ to define the multiplication, we obtain unital nonassociative algebras $S_f$ on the set of skew polynomials in $S[t;\sigma,\delta]$ of degree less than $m$. We study the structure of these algebras.When $S$ is a Galois ring and $f$ base irreducible, these algebras yield families of finite unital nonassociative rings $A$, whose set of (left or right) zero divisors has the form $pA$ for some prime $p$.For reducible $f$, the $S_f$ can be employed both to design linear $(f,\sigma,\delta)$-codes over unital rings and to study their behaviour.